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arXiv:2010.15518 · SciPost Physics Core

Bosonic and fermionic Gaussian states from Kähler structures

Lucas Hackl, Eugenio Bianchi A

Published Sep 22, 2021• 50 citations

Abstract

We show that bosonic and fermionic Gaussian states (also known as ``squeezed coherent states’’) can be uniquely characterized by their linear complex structure J J which is a linear map on the classical phase space. This extends conventional Gaussian methods based on covariance matrices and provides a unified framework to treat bosons and fermions simultaneously. Pure Gaussian states can be identified with the triple (G,\Omega,J) ( G , Ω , J ) of compatible Kähler structures, consisting of a positive definite metric G G , a symplectic form \Omega Ω and a linear complex structure J J with J^2=-\mathbb{1} J 2 = − 1 . Mixed Gaussian states can also be identified with such a triple, but with J^2\neq -\mathbb{1} J 2 ≠ − 1 . We apply these methods to show how computations involving Gaussian states can be reduced to algebraic operations of these objects, leading to many known and some unknown identities. We apply these methods to the study of (A) entanglement and complexity, (B) dynamics of stable systems, (C) dynamics of driven systems. From this, we compile a comprehensive list of mathematical structures and formulas to compare bosonic and fermionic Gaussian states side-by-side.

DOI: 10.21468/scipostphyscore.4.3.025

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We introduce a framework to identify where the total correlations and entanglement with a chosen degree of freedom reside within the rest of a system, in the context of bosonic many-body Gaussian quantum systems. Our results are organized into two main propositions. First, for pure Gaussian states, we show that every correlated mode possesses a unique single-degree-of-freedom partner that fully captures its correlations (consisting of entanglement), and we provide an explicit construction of this partner from the complex structure of the system's state. Second, for mixed Gaussian states, we constructively demonstrate that the notion of a partner subsystem splits into two: a correlation partner, which contains all classical and quantum correlations and need not correspond to a single degree of freedom, and an entanglement partner, which is always at most single-mode. Finally, we extend the construction of partners to multi-mode subsystems. Together, these results provide conceptual practical tools to study how bipartite correlations and entanglement are structured and where they can be found in complex Gaussian many-body systems.

Gaussian quantum states of bosonic systems are an important class of states. In particular, they play a key role in quantum optics as all processes generated by Hamiltonians up to second order in the field operators (i.e. linear optics and quadrature squeezing) preserve Gaussianity. A powerful approach to calculations and analysis of Gaussian states is using phase-space variables and symplectic transformations. The purpose of this note is to serve as a concise reference for performing phase-space calculations on Gaussian states. In particular, we list symplectic transformations for commonly used optical operations (displacements, beam splitters, squeezing), and formulae for tracing out modes, treating homodyne measurements, and computing fidelities.

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